One of the most important features of nonlinear dynamicai systems is that, as system
parameters are varied, qualitative changes in the overall behaviour of the system can
occur at a bifurcation . For smooth systems, the local bifurcations which occur under
the change of one parameter are well understood. Non-smooth dynamical systems,
which frequently arise due to the way certain physical processes are modelled, undergo
bifurcations which have not been widely studied. We examine a particular type of
bifurcation arising in a commonly occurring class of non-smooth dynamical system,
combining theoretical and experimental results.
In this thesis we are concerned with the study of the important class of dynamical
system we call impact oscillators, which undergo oscillations under the influence of
some forcing, and additionally can undergo impacts at rigid stops. Such systems are
of interest because a large number of physical and engineering systems display
behaviour which can be classified as impacting, where it is important to use a
dynamical analysis to identify and thus avoid the noise, wear or failure which could be
caused by repeated impacts producing unacceptably large loads. Recent interest in such
systems has concentrated on the unusual bifurcational behaviour which occurs when part
of an orbit begins to undergo low velocity impacts. Using analytical methods to locate
particular simple steady state solutions of an impact oscillator these grazing bjfurcations
are investigated. Comparisons are made between the behaviour of these special
bifurcations, which arise because of the instantaneous reversal of velocity in the
mathematical model of the impact process, and the standard bifurcations of smooth
dynamical systems.
An experimental study of an electromagnetically forced metal beam impacting against
a stop is used to show that the overall qualitative behaviour displayed by a simple
theoretical model is also displayed in a physical impact oscillator. Finally the
theoretical studies are related to a particular problem of offshore engineering and it is
shown how a very simple model can be used to explain some unusual observed
behaviour
